Uniform approximation of the Cox-Ingersoll-Ross process

Abstract

The Doss-Sussmann (DS) approach is used for uniform simulation of the CoxIngersoll-Ross (CIR) process. The DS formalism allows to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near zero. From a conceptual point of view the proposed method gives a better quality of approximation (from a path-wise point of view) than standard, or even exact simulation of the SDE at some discrete time grid.

Uniform approximation of the Cox–Ingersoll–Ross process via exact simulation at random times

“…The uniform approximation is connected with simulation of space-time bounded diffusions in fact (see [23] and [24,Chapter 5]). We note that the results of [22] were obtained under the restriction 4kλ > σ 2 . For the 4kλ ≤ σ 2 case, we did not succeed to extend the results of [22] in a Doss-Sussmann context.…”

“…We note that the results of [22] were obtained under the restriction 4kλ > σ 2 . For the 4kλ ≤ σ 2 case, we did not succeed to extend the results of [22] in a Doss-Sussmann context. In this paper we therefore follow an alternative approach.…”

“…In [22], we approximately constructed a generic trajectory of X(s) by simulating the firstpassage times of the increments of the Wiener process to the boundary of an interval and solving the ordinary differential equation after using the Doss-Sussmann transformation. Such a simulation is more simple than the one proposed in [9] and, moreover, has the advantage of a uniform nature.…”

“…In Section 4 we give some guidelines for numerical implementation of the proposed methods. In particular, we consider the method of Section 3, and exemplify in full detail the 4kλ = σ 2 case, which is at the border of applicability of the method in [22] in fact.…”

“…In [22], we considered uniform path-wise approximation of X(s) on an interval [t, t + T ] using the Doss-Sussmann transformation (see [27]) which allows for expressing any trajectory of X(s) by the solution of some ordinary differential equation that depends on the realization of w(s). The approximationX(s) in [22] is uniform in the sense that the path-wise error is uniformly bounded, i.e.…”